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Factoring Expressions
- Greatest Common Factor
(GCF)
- Grouping (4 terms)
Objectives
• I can factor expressions using
the Greatest Common Factor
Method (GCF)
• I can factor expressions using
the grouping method
Factoring?
• Factoring is a method to find the basic
numbers and variables that made up a
product.
• (Factor) x (Factor) = Product
• Some numbers are Prime, meaning they are
only divisible by themselves and 1
Method 1
• Greatest Common Factor (GCF) –
the greatest factor shared by two or
more numbers, monomials, or
polynomials
• ALWAYS try this factoring method
1st before any other method
• Divide Out the Biggest common
number/variable from each of the
terms
Greatest Common Factorsaka GCF’s
Find the GCF for each set of following numbers.
Find means tell what the terms have in common.Hint: list the factors and find the greatest match.
a) 2, 6
b) -25, -40
c) 6, 18
d) 16, 32
e) 3, 8
2
-5
6
161
No common factors?
GCF =1
Find the GCF for each set of following numbers.Hint: list the factors and find the greatest match.
a) x, x2
b) x2, x3
c) xy, x2y
d) 2x3, 8x2
e) 3x3, 6x2
f) 4x2, 5y3
x
x2
xy
2x2
Greatest Common Factorsaka GCF’s
3x2
1 No common factors?
GCF =1
Factor out the GCF for each polynomial:Factor out means you need the GCF times the
remaining parts.
a) 2x + 4y
b) 5a – 5b
c) 18x – 6y
d) 2m + 6mn
e) 5x2y – 10xy
2(x + 2y)
6(3x – y)
5(a – b)
5xy(x - 2)
2m(1 + 3n)
Greatest Common Factorsaka GCF’s
How can you check?
FACTORING by GCF
Take out the GCF EX:
15xy2 – 10x3y + 25xy3
How:
Find what is in common
in each term and put in
front. See what is left
over.
Check answer by
distributing out.
Solution:
5xy( )3y – 2x2 + 5y2
FACTORING
Take out the GCF EX:
2x4 – 8x3 + 4x2 – 6x
How:
Find what is in common
in each term and put in
front. See what is left
over.
Check answer by
distributing out.
Solution:
2x(x3 – 4x2 + 2x – 3)
Ex 1
•15x2 – 5x
•GCF = 5x
•5x(3x - 1)
Ex 2
•8x2 – x
•GCF = x
•x(8x - 1)
Factoring by Grouping
(When you have 4 terms or a higher even number)
Objective: After completing this section, students
should be able to factor polynomials by grouping.
Steps for factoring by grouping:
1. A polynomial must have 4 terms to factor by grouping.
2. We factor the first two terms and the second two terms
separately. Use the rules for GCF to factor these.
3 2. 2 2x x xex
3 2 2 2xx x 3 2 2
The GCF of
is .x x x
21 2x x
The GCF of
2 2 is 2.x
2
3. Finally, we factor out the "common factor" from both terms.
This means we write the 1 term in front and the 2 terms
left over, +2 , in a separate set of parentheses.
x
x
2 1x x 2 1x
Examples:
3 21. 6 9 4 6x x x 3 2 4 66 9 xx x
3 2 2
The GCF of
6 9 is 3 .x x x
The GCF of
4 6 is 2.x 23 2 3x x 2 2 3x
These two terms must be the same. 22 3 3 2x x
3 22. 1x x x 3 2 1xx x
3 2 2
The GCF of
is .x x x
The GCF of
1 is 1.x 2 1x x 1 1x
These two terms must be the same. 21 1x x
Examples:
3 23. 2 2x x x 3 2 22 xx x
3 2 2
The GCF of
2 is .x x x
The GCF of
2 is 1.x 2 2x x 1 2x
These two terms must be the same. 22 1x x
You must always check to see if the expression is factored completely. This
expression can still be factored using the rules for difference of two squares. (see 6.2)
22 1x x
2 1 1x x x
This is a difference of two squares.
Examples:
2 2 2 24. x y ay ab bx 2 2 2 2x y ay ab bx
2 2 2 2
The GCF of
is .x y ay y 2
The GCF of
is .ab bx b 2 2y x a 2b a x
These two terms must be the same.
You can rearrange the terms so that they are the same.
2 2y b x a
3 25. 2 2x x x 3 2 2 2xx x
3 2 2
The GCF of
is .x x x
The GCF of
2 2 is 2.x 2 1x x 2 1x
These two terms must be the same.
But they are not the same. So this
polynomial is not factorable.
Not Factorable
Try These:
Factor by grouping.
3 2
3 2
3 2
2
a. 8 2 12 3
b. 4 6 6 9
c. 1
d. 3 6 5 10
x x x
x x x
x x x
a b a ab
Solutions: If you did not get these answers, click the green
button next to the solution to see it worked out.
2
2
a. 4 1 2 3
b. 2 3 2 3
c. 1 1 1
d. 2 3 5
x x
x x
x x x
a b a
BACK
3 2a. 8 2 12 3x x x
3 2
2
2
8 2 12 3
3 4 12 4 1
4 1 2 3
x x x
xx x
x x
3 2 2
The GCF of
8 2 is 2 .x x x
The GCF of
12 3 is 3.x
BACK
3 2b. 4 6 6 9x x x
3 2
2
2
4 6 6 9
3 2 32 2 3
2 3 2 3
x x x
xx x
x x
3 2 2
The GCF of
4 6 is 2 .x x x
The GCF of
6 9 is 3.x
When you factor a negative out of
a positive, you will get a negative.
BACK
3 2c. 1x x x
3 2
2
2
1
1 11
1 1
1 1 1
x x x
xx x
x x
x x x
3 2 2
The GCF of
is .x x x
The GCF of
1 is 1.x
Now factor the difference of squares.
BACK
2d. 3 6 5 10a b a ab
23 6 5 10
3 2 5 2
2 3 5
a b a ab
a b a a b
a b a
The GCF of
3 6 is 3.a b 2
The GCF of
5 10 is 5 .a ab a
